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Correction to a Theorem on Total Positivity

  Published:2006-06-01
 Printed: Jun 2006
  • Carl Johan Ragnarsson
  • Wesley Wai Suen
  • David G. Wagner
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Abstract

A well-known theorem states that if $f(z)$ generates a PF$_r$ sequence then $1/f(-z)$ generates a PF$_r$ sequence. We give two counterexamples which show that this is not true, and give a correct version of the theorem. In the infinite limit the result is sound: if $f(z)$ generates a PF sequence then $1/f(-z)$ generates a PF sequence.
Keywords: total positivity, Toeplitz matrix, Pólya frequency sequence, skew Schur function total positivity, Toeplitz matrix, Pólya frequency sequence, skew Schur function
MSC Classifications: 15A48, 15A45, 15A57, 05E05 show english descriptions Positive matrices and their generalizations; cones of matrices
Miscellaneous inequalities involving matrices
Other types of matrices (Hermitian, skew-Hermitian, etc.)
Symmetric functions and generalizations
15A48 - Positive matrices and their generalizations; cones of matrices
15A45 - Miscellaneous inequalities involving matrices
15A57 - Other types of matrices (Hermitian, skew-Hermitian, etc.)
05E05 - Symmetric functions and generalizations
 

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